\chapter{Conversion of lattice units}
In lattice Boltzmann simulations, all the quantities used in the simulation are in lattice units. In order to convert parameters between physical and lattice units, the following reference scale must be defined: reference length $L_r$, reference density $\rho_r$ and reference velocity $u_r$. We use $L,\rho,t,\nu,c_s$ to represent length, density, time, viscosity and speed of sound in lattice units respectively, while $L',\rho',t',\nu',c_s'$ represent length, density, time, viscosity and speed of in real physical units. The reference units define the relation between the lattice units and physical unit:

\begin{eqnarray}
L_r=\frac{L'}{L},&\rho_r=\frac{\rho'}{\rho},&u_r=\frac{c'_s}{c_s}
\label{convert1}
\end{eqnarray}
 
In a simulation, $L,\rho,\nu,c_s$ are known, the real physical quantities $\rho',\nu',c_s'$ are also known according to measurements or literature. The only quantities that need to be determined are $L'$ and $L_r$. A dimensional analysis is conducted to determine the physical  and reference lengths. The Reynolds number in the simulation and the real physical problem should be the same because it is a dimensionless number. Therefore, we can obtain another equation as:

\begin{equation}
\frac{v L}{\nu}=Re=\frac{v' L'}{\nu'}
\label{convert2}
\end{equation}

Equation (\ref{convert2}) is equivalent to:
\begin{equation}
\frac{\nu'}{\nu}=\frac{v'L'}{vL}=L_r u_r
\label{convert3}
\end{equation}

Combining Equation (\ref{convert1}) and (\ref{convert3}), we obtain four equations to solve four unknowns: $\rho_r, u_r, L_r, L'$. The conversion in time can be carried out using  equation (\ref{convert4})

\begin{equation}
\frac{L_r}{u_r}=\frac{t'}{t}
\label{convert4}
\end{equation}

With $L_r, \rho_r, u_r$, it is possible to determine the real time interval $\Delta t'$, distance between nodes $\Delta x',\Delta y'$ and velocity $v'$.

 
